\section{Euler Angles}

Rotation around x-axis with $\Phi$
\begin{align*}
	T(\Phi) &= \begin{pmatrix}
		1 & 0 & 0 \\
		0 & \cos{\Phi} & \sin{\Phi} \\
		0 & -\sin{\Phi} & \cos{\Phi} 
	\end{pmatrix}
\end{align*}

Rotation around y-axis with $\Theta$
\begin{align*}
T(\Theta) &= \begin{pmatrix}
\cos{\Theta} & 0 &  -\sin{\Theta}\\
0 & 1 &  \\
\sin{\Theta} & 0 & \cos{\Theta} 
\end{pmatrix}
\end{align*}

Rotation around z-axis with $\Psi$
\begin{align*}
T(\Psi) &= \begin{pmatrix}
 \cos{\Psi} & \sin{\Psi} & 0 \\
-\sin{\Psi} & \cos{\Psi} & 0 \\
         0  &         0  & 1 
\end{pmatrix}
\end{align*}

\subsection{xyz-Convention}
Orthogonal transformation matrix from inertial to body frame
\begin{align}
	\underline{T}_i^b &= \begin{pmatrix}
		\ctheta \cdot \cpsi & \ctheta \cdot \spsi & - \stheta  
		\\
		\sphi\cdot \stheta \cdot \cpsi - \cphi \cdot \spsi & \sphi \cdot \stheta \cdot \spsi + \cphi \cdot \cpsi & \sphi \cdot \ctheta 
		\\
		\cphi \cdot \stheta \cdot \cpsi +\sphi \cdot \spsi & \cphi \cdot \stheta \cdot \spsi - \sphi \cdot \cpsi & \cphi \cdot \ctheta
	\end{pmatrix} \label{eq:Tib}
\intertext{Orthogonal transformation matrix from body to inertial frame}
	\underline{T}_b^i &= \begin{pmatrix}
		\ctheta \cdot \cpsi & \sphi\cdot \stheta \cdot \cpsi - \cphi \cdot \spsi & \cphi \cdot \stheta \cdot \cpsi +\sphi \cdot \spsi
		\\
		\ctheta \cdot \spsi & \sphi \cdot \stheta \cdot \spsi + \cphi \cdot \cpsi & \cphi \cdot \stheta \cdot \spsi - \sphi \cdot \cpsi
		\\
		- \stheta  & \sphi \cdot \ctheta  & \cphi \cdot \ctheta
	\end{pmatrix} \label{eq:Tbi}
\end{align}

Karman rotation matrix. 
\begin{align}
	\omega &= \underline{T}_2^b \cdot \begin{pmatrix} \dot{\Phi}  \\ 0 \\ 0 \end{pmatrix} + \underline{T}_1^b \cdot \begin{pmatrix} 0  \\ \dot{\Theta} \\ 0 \end{pmatrix} + \underline{T}_i^b \cdot \begin{pmatrix} 0  \\ 0 \\ \dot{\Psi} \end{pmatrix} 
	\\[1em]
	\omega &= \underline{V}_i^b \cdot \dot{\vec{\Phi}} \\[1em]
	\underline{V}_b^i &= \left(\underline{V}_i^b \right)^{-1}
	\\[2em]
	\underline{V}_i^b &= \begin{pmatrix}
		1 & 0 & -\stheta 	\\
		0 & \cphi & \sphi \cdot \ctheta 	\\
		0 & -\sphi & \cphi \cdot \ctheta
	\end{pmatrix} \label{eq:Vib}
	\\[2em]
	\underline{V}_b^i &= \begin{pmatrix}
		1 & \sphi \cdot t\Theta & \cphi \cdot t\Theta 	\\
		0 & \cphi & -\sphi 	\\
		0 & \frac{\sphi}{\ctheta} & \frac{\cphi}{\ctheta}
	\end{pmatrix} \label{eq:Vbi}
\end{align}
\clearpage